CHAPTER 5
IMPLEMENTATION OF INTEGRATED APPROACH FOR THE DEVELOPMENT OF INFINITE SOFTWARE

5.1 Assumptions

The implementation of the Taguchi method of design of experiments and finite element method for the robust design is based on the assumption that the effect due to each independent variable on the objective function is separable. This additive nature is valid only if there is no interaction between the independent variables. The software developed is meant only for additive models.

In addition to the above, the number of levels for all the design variables shall be the same. In case of mixed level models, a dummy level (s) has to be created by the user to make the number of variable levels same. Moreover, the INFINITE software can be used only for the single variable objective function.

5.2 Basic algorithm

When the INFINITE software is run, the main menu comes with four main command buttons viz. Read ANSYS File, Objective Function, Independent Variable and Taguchi Experiments. This is shown in Figure 5.1.

Figure 5.1 Opening menu of the INFINITE software

The user shall click the Read ANSYS File command button to start reading the ANSYS parametric language. Immediately a new Input ANSYS File window is opened to get the name of the input file. When the Objective Function command button is clicked, the user shall input the objective function parameters such as the type of problem (whether maximization or minimization) and the item to be optimized. The Independent Variable command button is meant for selecting the independent variables from the list of parameters defined in the ANSYS input file, specifying the number of levels and the level values. While the listing of parameters defined in the ANSYS file are automatically generated by the program, the selection of independent parameters, inputting the number of levels and level values are done by the user. Once the input has been specified by the user, the Taguchi Experiments button is clicked. The software automatically selects the right orthogonal array, sets the level values for each experiment, writes out the new ANSYS parametric file, conducts the experiments, retrieves the data and post process the experimental results.

The above steps are described below in detail.

5.2.1 Reading the ANSYS parametric file

Before conducting the design of experiments, the finite element model of the physical system to be analyzed shall be created using the ANSYS parametric language. A typical input file is given in Appendix B. This file contains all the independent design variables. When the INFINITE software is run, all the assigned variables defined in the ANSYS input file are read automatically. The program also reads the element type, material library (matlib in ANSYS) and defined materials and material properties.

Once the above variables are read, the program creates the unit instances of the defined parameter and the defined material unit class. This unit class stores all the information pertaining to the unit slot. The stored information can be used for further processing.

5.2.2 Total number of levels

Once the unit instances of all the assigned variables are stored, the graphical user interface permits the user to decide the number of levels of the experiment to be conducted for all the design variables. It is assumed that all the independent variables have same number of levels. Though in actual practice the number of levels of experiment for each independent variable may vary depending on the significance of each variable, this can be circumvented by creating dummy level(s) [6].

5.2.3 Selection of the independent variables

From the broad list of all the assigned variables, the user selects the independent design variables from the list as shown in Figure 5.2. After the selection of each independent variable, the program checks whether there are any dependent variable(s) which are functions of the selected independent variables. If there are any dependent variable(s), the program automatically and recursively disables the dependent variable(s) from further selection as an independent variable. This has been explained by means of the flowchart shown in Figure 5.3.

Figure 5.2 Selection of independent variables

Figure 5.3 Flowchart for the selection of independent variable

In case the selected independent variable is a function of variables, the program warns the selection of the user selected variable as an independent variable. However, if the user wants to really go ahead with the selection, then the function variables are considered as a constant and disabled from selecting as an independent variable.

In addition to the design variables, the materials and material properties also can be considered as independent variable. The program reads the material library and the defined material database. In case of material as level value, different materials are used for different levels. For material property as level value, different material property values are used for different levels without changing the material.

5.2.4 Selection of the objective function variable

The focus of the experiment is to study the behavior of the objective function variables such as deflection of beam, stress, heat flow etc., These objective function variables are the post processing variables known as state variables in ANSYS. Since the state variables changes depending on the element type, the program list only those post-processing items pertaining to the particular element type. Thus the system has been designed such that the list of available objective function variables is element-type context sensitive.

In order to select the objective function variable, the user shall click the type of objective function (minimization / maximization) and the variable parameters such as deflection, stress, etc. This has been shown in Figure 5.4

If the objective variable is a combination of many state variables, the user can define his/her own objective variable based on the state variables available in ANSYS. For example, if the aim of conducting design of experiment is to minimize the magnitude of displacement (d ) as given by equation 5.1, the user can do this by writing the equation in a post-processing file.

The post processing file can be the input for the selection of the objective function variable. A typical post processing file is given in Appendix C.

Figure 5.4 Selection of the objective function variable

5.2.5 Automatic selection of an orthogonal array

Once the user decides the design variables for conducting the experiments, the number of levels of each design variable and the objective function variable, the next stage is the selection of the right orthogonal array for the analysis. The orthogonal array unit class of the blackboard database contains the listing of 2, 3, 4 and 5 level standard orthogonal array tables. The slots of the unit class contains information on the maximum number of independent variables, the number of levels, number of experiments to be conducted and the level combination of each independent variable for all the experiments.

The selection procedure is based on the following criteria [3]

1. The selected orthogonal array shall have the same number of levels as selected by the user.

2. The maximum number of design variables of the selected orthogonal array shall be equal to or greater than the number of design variables selected by the user.

If the number of design variables selected is less than the permitted number of design variables as per orthogonal array, then the program automatically creates the dummy variables. These variables are not used for conducting the experiment, however it is essential to consider these variables for the calculation of error terms and interaction. Though the algorithm developed considers only the main effect of the independent variables, it is possible to check whether the interaction effects between the variables are significant or not based on the confirmation test and the percentage contribution due to error.

5.2.6 Conducting the experiment

The present method developed for the design of experiments simulates the real experiment. The finite element method is used as an analytical tool to conduct experiment. Each time a new experiment is conducted, the input parametric file of ANSYS is being modified. The modified input file contains the level values of the selected design parameters, materials, material properties, if any.

In addition to the modification of level values of design variables, the program appends a post-processing file which contains the objective function variable (say minimizing the bending stress of a beam). After writing out the new ANSYS parametric file, the ANSYS program is run. The ANSYS program stores the objective function values as a variable and writes out to a file. These objective function values also known as process parameters are available for further processing.

5.2.7 Analysis of the data

Once the experiments are conducted, the program automatically stores the process parameters and the corresponding experiment number and level combination of all the design variables in the blackboard. This raw data has been processed further to segregate the main effect of each individual variable. The following are the important parameters which the program automatically calculates.

i) Mean value of each level of a design variable

ii) Sum of square value of the design variables

iii) Total sum of square

iv) Percent contribution

v) Near optimal value of the objective function

vi) Confirmation test

vii) ANOVA (Analysis of Variance) test

The mean value of each level of a particular design variable is calculated by summing up the process parameters pertaining to an individual level. In case of L9 orthogonal array shown in Table 2.1, the mean value of level 2 for the variable 3 is given by the equation 5.2


Figure 5.5 Mean level value of design variables

It shall be noted that the grand mean of all the experiments is the same as the average of the mean values of each level of a design variable as shown in Figure 5.5. Based on the mean values of each design variable, the sensitivity analysis is performed.

The sum of square of individual design variable can be calculated using either of the following equations


where L is the number of level, N is the number of experiments conducted, R is the no of repetition per level which equals , T is the sum of process parameters of all the experiments, ......is the grand mean value of all the experiments which equals , and ...... is the mean value of jth level value of ith variable.

In case of L9 array which is given in Table 2.1, the total sum of square of variable 3 can be calculated using the equation 5.5 or 5.6.



Similarly the sum of square values for other variables can also be found.

The total sum of square (SSTO) is the sum of deviation of the experimental process parameters from the grand mean value of the experiment. This can be obtained from the equations 5.7 and 5.8.




where .... is the performance parameters for the kth experiment.

This total sum of square need not be the same as the total of sum of square of each individual variables. This is either due to the interaction effect between the design variables or due to the introduction of dummy variables, if any.

The percent contribution of each design variable is the ratio of the sum of squares of a particular design variable to the total sum of square of all the variables. This ratio indicates the influence of the design variable over the performance parameter due to the change in the level settings.

In order to find the near optimal value of the objective function, a new experiment is conducted by setting the near optimum level for each design variable. The near optimum level for any design variable can be easily found from the mean values of all the level. The optimum level values can be used as the initial value for further optimization problem.

It may be noted from the previous sections that the significance of individual design variables can be found from the percentage contribution. But it is not possible to categorically judge from the contribution value whether 5% contribution is significant or not. Using analysis of variance (ANOVA) approach, one can accept or reject a independent variable from the analysis given the confidence level, . This can be done by conducting F-test [1].

As per the F-test, a variable is significant only if the ratio of mean sum of square of a variable (MSV) to mean sum of square of error (MSE) is greater than the calculated F-value. The calculation of MSV and MSE is based on the accumulation method [1] as given by the following equations.



The calculated F-value is based on the statistical approach which obeys f-distribution with L-1 numerator degrees of freedom, N-L denominator degrees of freedom and as confidence level. The hypothesis for accepting or rejecting the significance of a variable is given by the following rules.

Null Hypothesis (Ho) : The design variable is not significant (5.11a)

Alternate Hypothesis (Ha) : The design variable is significant (5.11b)




It shall be noted that though the near optimal level combination is supposed to give the best performance, sometimes this may not happen due to the interaction effect. In that case a confirmation check may be needed. The near optimum performance parameter using confirmation test is based on the following formula. [1]


where NV is the number of design variables, ....... is the grand mean average of all the experiments, and ........ is the mean average value of the performance parameter for the near optimum level of i th variable.

However, if all the factorial effects were added together, it would be an over-estimate. Instead of giving an equal weight to all the factorial effects, weight is given to the design variables depending on the significance of each variable. This method is called the discount coefficient method [1]. As per this method the estimate value for the near optimum condition is given by the equation 5.13.



Here the significance coefficient is calculated based on the following formula



If the above calculated deviated significantly from the one calculated based on the optimum level values, the designer shall check the validity of assumptions made.

The following graphical outputs are available for easy visualization of the experimental results.

  1. Sensitivity analysis
  2. Percent contribution
  3. Optimal level combination

The sensitivity plot indicates the change in the performance parameter value due to the change in the level settings for each design variable. For easy comparison, the grand mean line also drawn. It may be noted that the mean level average value of each design variable is the same as the grand mean level. And hence the mean level values are centered around the grand mean value. If the mean values of all the levels of a design variable are very close to the grand mean line, the variable is not sensitive to changes in the level settings and hence is not significant. This is shown in Figure 6.5.

The percent contribution of each design variable is represented by means of a bar chart diagram as shown in Figure 6.6. From the plot one can deduce that a variable which is more sensitive to the level changes will contribute more.

The optimum level plot shows the optimum level value for each design variable and the optimum performance parameter value. This plot is combined with the orthogonal array table and the performance parameter values for each experiment. A typical plot is shown in Figure 6.7.